"Anger is not an emotion I feel very often; it takes a lot to piss me off," Liv Tyler is telling me. "But if I do get angry, I go into this whole other mode, and I almost black out."

The actress was recently required to channel that fury for an enraged, on-screen face-slapping incident. "I was terrified, because I didn't know what was going to happen," she confesses, her big blue eyes widening. "It's not something you can rehearse. Perhaps it's been lying dormant, like a dragon in the centre of the earth, because it's pretty wild when it comes out."

It's almost impossible to imagine 37-year-old Tyler, with her breathy, girlish voice and her other-worldly beauty, playing host to any sort of dragon, dormant or otherwise. But that's a common misconception, apparently. "I have a lot of opinions and I'm pretty bossy," she says.

We're in the living room of a hotel suite in Manhattan's SoHo and Tyler, having kicked off her very high heels, is lying on a sofa as we sip pink champagne. We are here to talk about The Leftovers, the 10-part drama that launches on Showcase Channel (Foxtel) on Tuesday and is based on a book by Tom Perrotta.

Tyler plays Meg, a troubled young woman dealing with a devastating loss ... but she's far from the only one. So is the rest of the town of Mapleton, and beyond, after the sudden, unexplained disappearance of two per cent of the world's population.

"This terrible event has happened, but it's three years later, and the story is not about the event itself," Tyler says. "It's about human beings and their journey and pain."

It's a television debut for Tyler, whose acting career so far has been confined to the big screen. In her teens, she starred in independent films including the comedy Empire Records and That Thing You Do! before making her name in Bernardo Bertolucci's dreamy, Tuscan-set Stealing Beauty. In her 20s, she was perfectly cast as Arwen, the Elf maiden in the Lord of the Rings trilogy, and appeared in other big-budget hits such as Armageddon and The Incredible Hulk. In her 30s, however, she has seemed to step back somewhat from the screen.
"Ever since Milo was born, I haven't been making many movies because I just can't imagine being away from home for long periods of time," she says. Milo, her son with her former husband Royston Langdon, of the British band Spacehog, is now nine, and The Leftovers films in and around New York City, meaning Tyler has been able to fulfil her domestic duties at home in Manhattan's West Village, as well as work.

"It has its challenges, too. There's something incredibly freeing about being 25 and getting on a plane to New Zealand, and being able to immerse yourself completely in your work," she says. Tyler sits up and shakes out her mane of dark hair. "Now, I get up and make Milo breakfast and walk him to school, then I have to go and beat somebody up at work, then figure out what's for dinner. It's the more grown-up version of being an actress."

Her decision to put her home life ahead of her career in recent years is a direct reaction to Tyler's own early experiences. "From a very young age I had this idea that if you are very successful in your career and you're giving all of your attention to that, then your family life ... possibly will not flourish as it might," she says.

Tyler was raised by her mother, the model and groupie Bebe Buell, who was the inspiration for Kate Hudson's character in Almost Famous and whose close friends included Mick Jagger and John Lennon. Famously, Tyler only discovered at 10 years of age that Aerosmith frontman Steven Tyler was her father, after meeting him backstage at a gig and noticing a resemblance between herself and his daughter, Mia. Until then she had believed her father to be Buell's on-off partner Todd Rundgren, also a musician. When Buell confirmed her daughter's suspicion – while Tyler was on stage – she recalls that the 10-year-old simply smiled and said: "Christmas is going to be really fun this year."

It was not the most ordered of childhoods, however. "My mum was very young [24] when she had me, and she didn't necessarily have all her tools in her toolbox," Tyler says. "We kind of grew up together, which was a beautiful thing. But we moved around a lot, too, between Maine and New York, and it was hard for me."

As for her newly acquired rock-star dad, she has said in the past that "I don't think he was in any position to be a father". When she was born, Steven Tyler was in the throes of his well-documented drug excesses. Rundgren was apparently aware of her paternity but, believing she needed a father, agreed to participate in the deception. "What an amazing thing Todd did for me. Luckily, it just made me feel more loved. I had two dads instead of one."

Tyler began modelling in adverts at 14 – although, she says, she loathed it. "I didn't like being told what to do. When I was 14, I had so many ideas, and I had to stand there, look pretty and not talk a lot. I didn't like that very much." For the past decade, having overcome her dislike of the job, she has been contracted to cosmetics giant Givenchy. Unusually for a model, she confesses to "always" being on a diet, but today looks svelte in skinny jeans, a peach blouse and white tuxedo jacket.

At 16, Tyler won one of her first acting roles courtesy of her father, alongside Alicia Silverstone in the video for Aerosmith's Crazy. Her first film role came a year later, in Silent Fall, followed by Empire Records. "I started really working right at the point when I probably would have become a f...-up," she says. "In those last three years of high school, you get really naughty and everything can go wrong. I'm grateful to my mother because, instead of that, I started travelling all over the world and working with incredible people, and I became very focused and disciplined. When everybody was doing acid and partying like crazy, I was at work on a movie in Tuscany."

That film – Stealing Beauty, in which she was cast by Bertolucci in the lead role alongside Jeremy Irons and Rachel Weisz – proved to be her breakthrough. Tyler is nothing short of rhapsodic when I mention it, saying it was a coming-of-age for her as much as for her character – the lush, ripe Lucy Harmon.
"Nothing has ever compared to the experience of making that film," she says with a sigh. "I graduated from high school the day before I went away, and turned 18 that summer. For my birthday, they roasted a pig for me, in the middle of a Tuscan farmhouse, with peacocks on the roof and bowls of cherries everywhere, and I was barefoot all the time."

Lucy travels to Tuscany desperate both to lose her virginity and to discover who her father is. Tyler has admitted that she attempted to separate herself from that particular storyline. "I tried my damnedest not to think of my own situation," she has said. "But at one point, after a take, I just started to cry."

Stealing Beauty premiered at Cannes, where Tyler's face was plastered across billboards around the town, catapulting her into the limelight. "When you start working at such a young age, everything is accelerated, and by the time you get to your mid-20s, you feel what most people probably feel in their mid-30s or early 40s," she says. She married Langdon at 25, and gave birth to Milo a year later. "In my 20s, I definitely had a sense of wanting to create stability for myself."

The couple divorced in 2008, after five years of marriage, but Tyler is impressively sanguine about it. "The idea that we can share a roof with someone else forever is a beautiful notion, but it's much harder to actually do. I grew up in a very bohemian environment, and I always knew that it doesn't mean you don't love that person, or that they're not important to you," she says. "But I'm completely devoted to Milo and our little family, no matter how eccentric it might seem to others."

In fact, Tyler has spent a lot of time with Langdon's family in Leeds, in the north of England. "We had the best Yorkshire Christmas last year, it was so under the radar," she says. "And I'm so happy that Milo gets to have that in his foundation – the humour and the groundedness, those northern qualities."

These days, she is close to both Rundgren and Steven Tyler and sees them regularly, as does Milo. She also believes strongly in "the family you create for yourself", which in her case means a close network of girlfriends including Kate Hudson, Gwyneth Paltrow and Stella McCartney.

I ask whether she was attracted to the spiritual dimension of The Leftovers – the "sudden departure" has been compared to the notion that on Judgment Day some people will remain on Earth, while others ascend to heaven. "I'm spiritual, yes, but I don't think about Judgment Day," she says.

She does admit, however, to thinking about having more offspring. "I hope I will have more children," she says. "I'm 100 per cent planning on it. If the stork could just drop it off on my roof, I'd be so happy – I'd have, like, 20." (Indeed, at the time of going to press, her agent confirmed that Tyler was pregnant to her boyfriend Dave Gardner, with Steven Tyler commenting he is "over the moon" at the news.)

Gardner and Tyler have been seeing each other for some time, but whether Tyler will marry again is uncertain. "In your 30s, you're not in that stage any more of princes and happily ever after. It's a different stage of acceptance about the realities of love and relationships. Forever is a long time."